A luminous object is separated from a screen by distance d. A convex l...
Understanding the Problem
In this scenario, we have a luminous object and a screen separated by a distance d. A convex lens is placed between them, forming a clear image on the screen. The goal is to determine the maximum possible focal length of this lens.
Lens Formula and Image Formation
The lens formula is given by:
1/f = 1/v - 1/u
Where:
- f = focal length of the lens
- v = image distance from the lens
- u = object distance from the lens
When the lens is positioned optimally between the object and the screen, we can maximize the focal length.
Positioning the Lens
1. Maximum Focal Length:
- For a distinct image on the screen, the object (O), lens (L), and image (I) must satisfy the condition O - L + L - I = d.
- The maximum separation occurs when the lens is at the midpoint of the total distance (d).
2. Object and Image Distances:
- Let u be the object distance from the lens and v the image distance from the lens.
- At maximum focal length, u + v = d.
Calculating Focal Length
1. Setting the Distances:
- If we set u = d/2 and v = d/2, we can substitute these values into the lens formula.
- The maximum focal length f occurs when the distances are equal, yielding the result.
2. Result:
- After calculations, it is found that the maximum possible focal length of the lens is d/4.
Conclusion
Thus, the maximum possible focal length of the convex lens, allowing it to form a distinct image on the screen placed at distance d from the object, is indeed d/4. This aligns with option 'D'.
A luminous object is separated from a screen by distance d. A convex l...
From lens displacement method, fmax = d/4